What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.
AAAaannndd the bot gets it wrong yet again!
√7 ≈ 2.645 and 4/7 < 2.645 < 5/7
√8 ≈ 2.828 and 5/7 < 2.828 < 6/7
so √7 < 19/7 < √8
Apologies for the previous incorrect response. You are absolutely right.
To find a rational number between \(7 - \sqrt{2}\) and \(8 - \sqrt{2}\) with a denominator of 7, we can compare these numbers to the rational numbers with a denominator of 7:
\(\sqrt{7} \approx 2.645\) and \(\sqrt{8} \approx 2.828\).
We know that \(\frac{4}{7} < 2.645 < \frac{5}{7}\) and \(\frac{5}{7} < 2.828 < \frac{6}{7}\).
Therefore, a rational number with a denominator of 7 that lies between \(7 - \sqrt{2}\) and \(8 - \sqrt{2}\) is \(\frac{19}{7}\).
So, the answer is \(\frac{19}{7}\).
Describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole numbers? Are all whole numbers also rational numbers? Give examples to support your answer.(4 points)
To find a rational number with a denominator of 7 that is between 7–√ and 8–√, we need to express both numbers with a denominator of 7.
First, we need to simplify the given expressions:
7–√ = 7 – 1√ = 7 – √1
8–√ = 8 – 1√ = 8 – √1
Next, we need to rationalize the denominators:
7–√ = 7 – √1 = 7 – √1 * (√1/√1) = 7 – √1 * (√1/√1) = 7 – (√1/√1) = 7 – (√1/1) = 7 – (√1) = 7 – 1/1 = 7 – 1 = 6
8–√ = 8 – √1 = 8 – √1 * (√1/√1) = 8 – √1 * (√1/√1) = 8 – (√1/√1) = 8 – (√1/1) = 8 – (√1) = 8 – 1/1 = 8 – 1 = 7
So, we have 6/7 as the fraction between 7–√ and 8–√.
To find a rational number between 7–√ and 8–√, we first need to simplify the expressions 7–√ and 8–√.
1. Start by noting that the square root of any number is always positive, so √x is greater than 0 for any positive x.
2. Simplify 7–√ by noting that it is equivalent to 7 + (-√1). Since the square root of 1 is 1, this simplifies to 7 – 1 = 6.
3. Simplify 8–√ by noting that it is equivalent to 8 + (-√1), which simplifies to 8 – 1 = 7.
Now we have the range for the rational number: 6 < x < 7.
Since we are looking for a rational number with a denominator of 7, we need to find a number that can be expressed as `a/7` where `a` is an integer.
To find a suitable number for `a`, we can consider the mid-point between 6 and 7, which is 6.5. Multiplying this by 7 gives us a value of 45.5.
However, since we need the number to be an integer, we need to consider two possible fractional values: one slightly less than 45.5 and one slightly greater.
To find these values, let's examine the nearest whole numbers around 45.5:
45 < 45.5 < 46
So, the two possible values for `a` are 45 and 46.
Now we can write these two numbers as improper fractions:
For 45, we have `45/7`.
For 46, we have `46/7`.
Therefore, the rational numbers with a denominator of 7 that are between 7–√ and 8–√ are:
45/7 and 46/7.
To find a rational number between \(7-\sqrt{2}\) and \(8-\sqrt{2}\) with a denominator of 7, we can first express \(7-\sqrt{2}\) and \(8-\sqrt{2}\) with a common denominator of 7.
\[
7-\sqrt{2} = \frac{7}{1} - \frac{\sqrt{2}}{1} = \frac{7}{1} - \frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{7}{1} - \frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{1} = \frac{7}{1} - \frac{\sqrt{4}}{1} = \frac{7}{1} - \frac{2}{1} = \frac{5}{1}
\]
\[
8-\sqrt{2} = \frac{8}{1} - \frac{\sqrt{2}}{1} = \frac{8}{1} - \frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{8}{1} - \frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{1} = \frac{8}{1} - \frac{\sqrt{4}}{1} = \frac{8}{1} - \frac{2}{1} = \frac{6}{1}
\]
So, we need to find a rational number between \(\frac{5}{1}\) and \(\frac{6}{1}\) with a denominator of 7. To find such a number, we can take the average of \(\frac{5}{1}\) and \(\frac{6}{1}\) which gives us \(\frac{5+6}{2} = \frac{11}{2}\).
To write this as an improper fraction, we divide the numerator 11 by the denominator 2, which gives us:
\[
\frac{11}{2} = \boxed{ \frac{11}{2}}
\]